Solution :

Suppose that the sequence {a_n} contains only finitely many composite numbers. That means that there exists N such that for all n > N a_n is odd.

Consider the binary representation of a_n, n > N. The last digit of a_n must be 1, which in turn implies that the fractional part of the binary representation of  has all its digits equal to 1 after N-th position. This contradicts the irrationality of .